Betti categories of graded modules and applications to monomial ideals and toric rings

TL;DR

This paper introduces Betti categories for graded modules, providing a unified categorical framework that determines minimal free resolutions for monomial ideals and toric rings, with new combinatorial tools and analogs.

Contribution

It develops the concept of Betti categories for modules over graded polynomial rings and small categories, unifying the treatment of monomial ideals and toric rings.

Findings

Betti category is a finite combinatorial object that determines minimal free resolutions.

For monomial ideals, Betti category coincides with the Betti poset.

Provides an analog of the lcm-lattice for toric rings.

Abstract

We introduce the notion of Betti category for graded modules over suitably graded polynomial rings, and more generally for modules over certain small categories. Our categorical approach allows us to treat simultaneously many important cases, such as monomial ideals and toric rings. We prove that in these cases the Betti category is a finite combinatorial object that completely determines the structure of the minimal free resolution. For monomial ideals, the Betti category is the same as the Betti poset that we studied in a previous article. We describe in detail and with examples how the theory applies to the toric case, and provide an analog for toric rings of the lcm-lattice for monomial ideals.